Let X be a set containing 3 elements. If two subsets A and B of X are picked at random, the probability that A and B have the same number of elements is k then 1/k is______.

Answer

To solve the problem, we need to determine the probability that two randomly chosen subsets \( A \) and \( B \) of a set \( X \) with 3 elements have the same number of elements. Let's denote the set \( X = \{E_1, E_2, E_3\} \).

### Step 1: Determine the total number of subsets of \( X \)
The total number of subsets of a set with \( n \) elements is given by \( 2^n \). Here, \( n = 3 \).

\[ \text{Total subsets} = 2^3 = 8 \]

### Step 2: List all possible subsets of \( X \)
The subsets of \( X \) are:
1. \( \emptyset \) (0 elements)
2. \( \{E_1\} \) (1 element)
3. \( \{E_2\} \) (1 element)
4. \( \{E_3\} \) (1 element)
5. \( \{E_1, E_2\} \) (2 elements)
6. \( \{E_1, E_3\} \) (2 elements)
7. \( \{E_2, E_3\} \) (2 elements)
8. \( \{E_1, E_2, E_3\} \) (3 elements)

### Step 3: Determine the number of ways to choose subsets \( A \) and \( B \)
Since both \( A \) and \( B \) can be any of the 8 subsets, the total number of ways to choose \( A \) and \( B \) is:

\[ \text{Total ways to choose } A \text{ and } B = 8 \times 8 = 64 \]

### Step 4: Count the favorable outcomes where \( A \) and \( B \) have the same number of elements
We need to count the cases where both subsets have the same number of elements. The possible sizes for \( A \) and \( B \) are 0, 1, 2, or 3.

- **Both have 0 elements**: \( \emptyset \) (1 way)
- **Both have 1 element**: Choose 1 element from 3, so \( \binom{3}{1} = 3 \). Thus, \( 3 \times 3 = 9 \) ways.
- **Both have 2 elements**: Choose 2 elements from 3, so \( \binom{3}{2} = 3 \). Thus, \( 3 \times 3 = 9 \) ways.
- **Both have 3 elements**: \( \{E_1, E_2, E_3\} \) (1 way)

Now, summing these favorable outcomes:

\[ 1 + 9 + 9 + 1 = 20 \]

### Step 5: Calculate the probability \( k \)
The probability \( k \) that subsets \( A \) and \( B \) have the same number of elements is given by the ratio of favorable outcomes to total outcomes:

\[ k = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{20}{64} = \frac{5}{16} \]

### Step 6: Find \( \frac{1}{k} \)
Now, we need to find \( \frac{1}{k} \):

\[ \frac{1}{k} = \frac{16}{5} \]

Thus, the final answer is:

\[ \frac{1}{k} = \frac{16}{5} \]